shaft-design

name	shaft-design
description	Complete shaft design — combined bending+torsion, DE-Goodman/ASME-Elliptic criteria, critical speed, deflection, keyway stress concentration, bearing seat sizing. Shigley Ch.6-7.
metadata	{"priority":9,"promptSignals":{"phrases":["shaft","shaft design","torque transmission","critical speed","shaft diameter","power transmission shaft"],"minScore":4}}

Shaft Design — Complete Skill

Step 1: Load Analysis

• Identify all torques (T) — from gears, pulleys, couplings
• Identify all transverse loads (F) — gear forces, pulley tensions
• Build free-body diagram in both planes (horizontal + vertical)
• Compute M_H and M_V at each critical section
• Resultant moment: M = √(M_H² + M_V²)

Step 2: Gear/Pulley Forces

Spur gear: W_t = 33000·HP/V (Imperial), W_t = T/r (SI) W_r = W_t·tan(φ) (φ = pressure angle, typically 20°) W_a = 0 (no axial for spur)

Helical gear: W_t = T/r, W_r = W_t·tan(φ_n)/cos(ψ), W_a = W_t·tan(ψ)

Belt drive: T = (F₁-F₂)·r; F₁/F₂ = e^(μθ) Net radial force on shaft: F = F₁ + F₂ (for V-belt: multiply by 0.9)

Chain: Similar to belt but no slip; F = T/r (tight side only, slack = 0 assumed)

Step 3: Critical Section

Locate where M and T are simultaneously large — usually at gear/bearing junction. Check: shoulders, keyways, press-fit locations (stress concentration sites).

Step 4: Stress Concentration Factors

Feature	Kt (bending)	Kts (torsion)
Shoulder fillet r/d=0.02	2.7	2.2
Shoulder fillet r/d=0.1	1.7	1.5
Keyway (end milled)	2.14	3.0
Press fit	2.0-3.0	—
Transverse hole	2.0-2.4	1.5-2.0

Fatigue factors: Kf = 1 + q(Kt-1), Kfs = 1 + qs(Kts-1)

Step 5: DE-Goodman Criterion (recommended)

Fully reversed bending + steady torsion (most common shaft loading):

σ_a = Kf · 32M / (πd³) σ_m = 0 (bending is fully reversed) τ_a = 0 τ_m = Kfs · 16T / (πd³)

Von Mises: σ'_a = √(σ_a² + 3τ_a²) = Kf · 32M / (πd³) σ'_m = √(σ_m² + 3τ_m²) = Kfs · 16T√3 / (πd³)

Goodman safety factor: 1/n = σ'_a/Se + σ'_m/Sut

Required diameter (DE-Goodman): d = [16n/π · (√(4(KfM)²+3(KfsT)²)/Se + √(4(KfM)²+3(KfsT)²)/Sut·(terms))]^(1/3)

Simplified for fully reversed bending + mean torsion: d³ = 16n/π · [4(KfMa)²+3(KfsT)²]^0.5/Se + [4(KfMm)²+3(KfsT)²]^0.5/Sut

Step 6: ASME-Elliptic Criterion (alternative)

1/n² = (σ'_a/Se)² + (σ'_m/Sy)²

Step 7: Static Check (first cycle yield)

von Mises stress: σ_von = √(σ² + 3τ²) σ_max = 32M/πd³ + P/A (add axial if present) τ_max = 16T/πd³ + VQ/Ib (add transverse shear if significant) Check: σ_von < Sy (usually not governing if fatigue check passes)

Step 8: Critical Speed (Rayleigh-Ritz)

Single mass: n_c = π/2 · √(g/δ) rpm, δ = static deflection under own weight + loads

Multi-mass (Dunkerley's equation): 1/n_c² = 1/n₁² + 1/n₂² + ... + 1/n_shaft²

Where n_i = critical speed if only mass i were present.

Rule: Operating speed < 0.75·n_c (subcritical) OR > 1.4·n_c (supercritical — pass through quickly)

Shaft alone (simply supported): n_c = π²/L² · √(EIg/w) rpm (w = weight per unit length)

Step 9: Deflection Check

At bearing locations: δ < 0.001in/ft·L (bearing misalignment limit) At gears: δ < 0.005" (spur), slope < 0.0005 rad (helical) At seals: slope < 0.001 rad

Step 10: Keyway Design

Key width w = d/4, height h = d/6 (standard proportions) Key length L: shear of key: τ = 2T/(dLw) < Ssy Bearing on key/hub: σ = 4T/(dLh) < 0.9Sy

Minimum Shaft Diameter Selection (AGMA 9002)

d_min from: torque capacity, bending capacity, critical speed, stiffness — take the maximum.

Output

Provide: d (required), d_selected (next standard size), n_fatigue, n_yield, n_c, governing failure mode.